INCESSANT TRANSMISSION 9/6 



He can deduce that if it is in water it will not stay there, for W^ W 

 has probability zero, but will go usually to the bank, for W-^ B 

 has the highest probability in the column. From the bank it will 

 probably go to the water, and then back to the bank. If under a 

 pebble it also tends to go to the water. So clearly it spends much 

 of its time oscillating between bank and water. Time spent under 

 the pebbles will be small. The protocol given, which was constructed 

 with a table of random numbers, shows these properties. 



Thus the matrix contains information about any particular 

 system's probable behaviour. 



Ex. 1 : Had the P-column of the matrix a 1 in the lowest cell and zero elsewhere, 



what could be deduced about the insect's mode of life ? 

 Ex. 2: A fly wanders round a room between positions A, B, C, and D, with 



transition probabilities: 



I 



A B C D 



One of the positions is an unpleasantly hot stove and another is a fly-paper. 

 Which are they? 



Ex. 3: If the protocol and matrix of Ex. 9/4/1 are regarded as codings of each 

 other, which is the direction of coding that loses information ? 



9/6. Equilibrium in a Markov chain. Suppose now that large 

 numbers of such insects live in the same pond, and that each behaves 

 independently of the others. As we draw back from the pond the 

 individual insects will gradually disappear from view, and all we will 

 see are three grey clouds, three populations, one on the bank, one 

 in the water, and one under the pebbles. These three populations 

 now become three quantities that can change with time. If they 

 are dg, dpf,, and dp respectively at some moment, then their values 

 at one interval later, d^' etc., can be found by considering what their 

 constituent individuals will do. Thus, of the insects in the water, 

 three-quarters will change over to B, and will add their number on 

 to dg, while a quarter will add their number to dp. Thus, after the 

 change the new population on the bank, dg', will be idg+^dyy+^dp. 

 In general therefore the three populations will change in accordance 

 with the transformation (on the vector with three components) 

 It must be noticed, as fundamentally important, that the system 



r] ' — ^d 4- ^d 4- ^d 

 d '— ^d A- i-d 



"fy — ^^Ug -f 4.Up 



dp = idw + ^dp 



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